A Stirling cycle heat engine generally consists of a variable volume sealed enclosure which contains a fixed quantity of a well behaved relatively ideal gas, typically helium. The enclosure is subdivided into two compartments by a gas permeable barrier which thermally isolates the two compartments from each other. The barrier is a porous medium made of very finely divided material which stores heat to and from gas passing through it.
The two thermally isolated compartments are separately held in close thermal contact with two thermal reservoirs at differing temperatures. Means are provided to vary the allocation of gas between the two compartments in a timed, phase shifted relationship with respect to the cyclic variation of the overall enclosure volume. The gas pressure is always very close to equal in both compartments because the thermal barrier between them is freely gas permeable. The phase shift causes a preponderance of gas to reside in one compartment during the portion of the cycle when the overall volume is being reduced or "compressed", and in the other compartment when the overall volume is being increased or "expanded". As a result, net heat is rejected to an attached thermal reservoir by the compartment which contains the preponderance of the gas during compression, and net heat is absorbed from another thermal reservoir by the compartment which contains the preponderance of the gas during expansion.
If the compartment which contains the dominant quantity of gas during expansion is maintained at an elevated temperature with respect to the temperature of the compartment containing the predominance of gas during compression, net work is available from the cycle; it produces power. But if compression occurs when the predominance of gas resides in the warmer compartment and expansion when the bulk of the gas resides in the cooler one, external work will be required to carry out the overall volume variation, and a quantity of heat will be pumped from the cooler reservoir to the warmer; the cycle produces refrigeration.
Stirling and Ericsson engines are potentially useful as both engines and refrigerators, if competitive efficiencies are attained. Present vapor cycle refrigerators reach about 75% of Carnot equivalent coefficient of performance.
The gas permeable medium which thermally isolates the two compartments first stores and then gives back a certain quantity of heat from the gas. At the end of the cycle the medium must be returned to the identical state in which it started the cycle. In a Stirling cycle heat engine, the gas is made to transit the porous medium "regenerator" at constant volume. The pressure of the gas within the overall volume thus changes as gas is shifted through the regenerator. In an Ericsson cycle heat engine, the gas is made to transit the regenerator at constant pressure. The cycles differ crucially in the behavior of the overall volume during the shifting of the gas through the regenerator. A Stirling engine does the shifts isochorically whereas an Ericsson engine does them isobarically.
The Ericsson cycle also differs from the Stirling in the allocation of displacement between the two compartments within the overall volume. Stirling engines usually have approximately equal displacements in both compartments. In Ericsson engines, the volume variation of the warmer compartment is made greater than the volume variation of the cooler one by the ratio of design absolute temperatures at which the two compartments will operate. In an Ericsson engine the expansion pressure ratio must always equal the compression pressure ratio.
In usual practice, Stirling engines do not perform the separate compression, shift, expansion, and shift back steps in an isolated and discrete manner. Rather, each step overlaps it neighbors. In theory, the effectiveness of the cycle depends only on obtaining a net predominance of gas in one compartment during the compression process and in the other compartment during the expansion process. During some part of each cycle a certain amount of heat is transferred in the wrong direction in each of the two compartments, but the preponderance of the gas resides in the other compartment during such times.
The Ericsson cycle must necessarily isolate the cycle into its four discrete steps. Thus, most prior Ericsson cycle engines have employed continuous flow compression and expansion processes, and counter-flow recuperators rather than regenerators. But it is also possible to operate a two compartment engine resembling a Stirling in the Ericsson cycle by, for instance, actuating the pistons through properly shaped and timed cams in conjunction with the volume variation ratios earlier described.
Like a Stirling, a positive displacement Ericsson cycle engine must maintain a preponderance of gas in one compartment during the overall volume compression, and in the other compartment during overall volume expansion. Similarly, there is a quantity of reverse heat transfer in each compartment during some portion of the cycle because the pressure change acting on the gas remaining within induces temperature change and heat flow during the opposing step of the cycle.
Stirling and related engines of the prior art have generally employed pistons, cylinders and a variety of linkages to obtain the necessary phased relationships of thermally isolated volumes connected through a regenerator. The pistons have been connected to cranks, swash plates, cams, etc. Engines have been constructed with single and multiple pistons, single and dual action pistons, free pistons, and fixed and moving regenerators. The Wankel displacement mechanism originally developed for Otto cycle engines has also been adapted to the Stirling cycle. Generally, the variable volume actuation means produce sinusoidal volume variations.
As earlier mentioned, there is always a certain amount of heat flow in the reverse direction in both compartments during some portions of the Stirling and Ericsson cycles. It is instructive to estimate the magnitude of forward, reverse and net heat flows in typical engines.
Because the volumes vary sinusoidally and the cycle is not isolated into truly discrete steps, the Stirling cycle can be difficult to model mathematically. To allow calculation of the reversing heat flows the cycle must be treated as a unified whole which includes the effects of the overlapping of steps.
With assumption of isothermal compression and expansion and neglect of detail regenerator effects it is easy to derive a model which explicitly treats the overlapping sinusoidal volume variations. The key simplification is the presumption that the gas temperatures in each compartment always equal the attached thermal reservoir temperatures. In other words, the heat exchangers are presumed to be perfect. Pressure becomes a function of the volumes and temperatures only, allowing the straightforward derivation of two integrals which represent the work done and heat transferred in each of the two compartments. The two integrals do not have an easy analytical solution, but they easily integrate numerically.
A computer spreadsheet program may be used to perform such numerical integrations. The cycle is divided into discreet intervals and the integral terms derived by the above method evaluated for each. The intervals are summed to obtain the net cycle work and heat flows.
The use of spreadsheet software allows quick trial of varying parameters and observance of the effects.
These methods were used to estimate the magnitudes of forward and reverse heat flows in a Stirling machine with working gas temperatures such as would be encountered in a deep freezer refrigeration application; 225.degree. K. gas temperature in the heat absorbing compartment and 325.degree. K. in the heat rejecting compartment. Live volume in each compartment was set equal to dead volume and the two compartments were assumed to have the same total variation, as is typical of most prior art Stirling engine designs. The phase angle was chosen to maximize specific work. The cycle was divided up into eight intervals of integration and the heat flow calculated for each.
Like any ideal analysis of the Stirling cycle, the method predicts Carnot equivalent performance. But inspection of the heat flow terms for the intervals comprising the cycle reveals the principle efficiency bottleneck. The forward heat flows are only about 25% larger than the reverse heat flows. To pump one net watt the machine will push five and pull back four. The total heat flow exceeds the net heat flow by almost an order of magnitude.
The reverse heat flow is also huge compared to the net mechanical work of the cycle. Clearly an inefficiency in the heat transfer process will profoundly affect cycle efficiency. The ideal cycle model predicts Carnot equivalent performance only because the heat exchangers were presumed to be perfect. Real heat exchangers are far from perfect, as real Stirling engines are far from efficient. Imperfect heat transfer allows the gas temperature to fluctuate. This will cause a "hysteresis loss" in the heat transfer process.
The magnitude of hysteresis loss in the heat exchangers can be estimated after making reasonable assumptions as to the approach temperatures attainable in the heat exchangers. Obviously, the gas temperature must differ from the heat exchanger surface temperature to drive heat flow. Thus, the gas temperature in both heat exchangers will exceed each heat exchanger surface temperature during compression or overall volume reduction. Likewise, the gas temperature in both compartments will be lower than the respective heat exchanger surface temperature during expansion or overall volume increase. Additionally, some of the pressure variation will occur semi-adiabatically as the temperature of the gas fluctuates.
The effect of the approach temperature fluctuations is to increase the amount of work required to compress the gas in the expansion compartment during overall volume reduction, and reduce the work available from the gas in the compression compartment during the overall volume increase. Both these effects increase the work required to perform the refrigeration cycle, or reduce the net work available from an engine cycle.
The magnitude of hysteresis loss is directly related to the magnitude of the temperature fluctuation and the quantity of gas in the adverse compartment. It is easily estimated by retaining the assumptions that compression and expansion are isothermal, but occur at temperatures above and below the actual heat exchanger surface temperatures. Then, the work and heat transfers in the reverse directions are found by scaling up the ideal reverse direction heat flows from the ideal cycle model by the ratio of the absolute gas temperatures during the compression and expansion steps in the respective compartments. The difference between scaled and ideal values is the hysteresis loss.
Generally, it is extremely difficult to hold the gas temperature fluctuation below 20.degree. K. (10.degree. K. above and below the heat exchanger inner surface temperature) in either of the two compartments. If the refrigeration machine described above had this fluctuation the compression work expended on the expansion chamber would be increased by the ratio 245/225 or 1.09. Similarly the expansion work available from the gas in the compression compartment would be only 305/325 or 0.94 times the ideal. Because the reverse direction work and heat flows are so great, these effects combine to seriously degrade efficiency. Under the example conditions the coefficient of performance is reduced to less than unity by heat exchanger hysteresis loss, as compared to the Carnot equivalent COP of 2.25. The machine could not work as an engine.
This analysis is very much oversimplified by its failure to treat the semi-adiabatic portions of the cycles. Even so, it predicts losses in the range of those experienced in practice.
Clearly, heat exchanger hysteresis loss is significant. Its magnitude is at least partly decreased by reducing the dead volume of the two compartments to lessen the quantity of gas in the wrong compartment during the adverse portion of the cycle. However, even at the practical minimum of dead volume ratio, about 10%, forward heat transfer exceeds the reverse by only a ratio of two. The cycle remains quite sensitive to temperature fluctuation in the heat exchangers.
Unfortunately, reducing the dead volume greatly increases the cycle pressure ratio. This is undesirable because of adverse effects on the performance of the regenerator. It is useless to limit the cycle pressure ratio by making the phase shift unconventionally large because the relative magnitude of the reversed heat flows will also be increased by this method.
Generally, detailed analysis of the events in the regenerator has been neglected due to the great difficulty of modeling the unsteady, reversing flows and varying pressures. It is usually presumed that the regenerator is compatible with the isochoric shifting of the gas from one compartment to the other, and with the inevitable cycle pressure variation. This presumption is not true. The pressure fluctuations induced during the isochoric shifts and the cycle pressure fluctuations both cause the gas to go out of thermal equilibrium with the porous regenerator medium throughout its length. Unfortunately, packed bed or porous regenerators are efficient only if they remain in close thermal equilibrium with the gas. They work best under isobaric conditions. Heat exchange with the gas ideally occurs only in a well defined sharp front that is displaced in proportion to gas movement through the medium. The spurious heat exchange induced by the isochoric pressure fluctuation involves irreversibilities which are very damaging the cycle efficiency.
The Ericsson cycle, with its isobaric shifting of the gas through the regenerator, avoids irreversibility losses during the shift steps. It is thus more compatible with porous medium regeneration than the Stirling cycle. Both cycles must have pressure variation during the cycle in order to work at all, but the pressure ratio should be limited to minimize the extent of disequilibrium in the regenerator. Thermal lag in the heat exchange between the gas in the regenerator and the regenerator medium increases the work required to execute the refrigeration cycle or reduces the output of an engine.
Gas movement through the regenerator during pressure changes also drives a heat leak of potentially significant magnitude. Both compression and expansion tend to push the temperature front in the regenerator towards the cooler compartment. This movement of the front occurs on top of the normal front movements induced by the intrinsic cycle shifts. Unfortunately the pressure induced front movements add rather than cancel. There is thus always some unwanted heat transfer from warm compartment to cool, even in regenerators that should be very nearly perfect. Low cycle pressure variation reduces but never eliminates the effect.
Between them, heat exchanger hysteresis loss and regenerator compatibility problems account for the limited efficiency attained so far in Stirling cycle engines and refrigerators. The former alone is quantifiably so great as to preclude the possibility of constructing sinusoidal Stirling cycle refrigerators efficient enough to compete with vapor cycle machines. To closely approach the Carnot efficiency limit one must employ the method described hereinafter to control the magnitudes of heat exchanger hysteresis loss and operate in a manner more compatible with porous media regenerators.